**B**ased on a (basic) question on X validated, I re-read Halmos‘ (1946) famous paper on the non-existence of centred moments of order larger than the sample size. While the exposition may sound a wee bit daunting, the reasoning is essentially based on a recursion and the binomial theorem, since expanding the kth power leads to lesser moments, all of which can be estimated from a subsample.

## Archive for unbiasedness

## re-reading Halmos (1946)

Posted in Books, Kids, Statistics, University life with tags central moments, cross validated, Paul Halmos, unbiasedness on May 16, 2020 by xi'an## unbiased MCMC discussed at the RSS tomorrow night

Posted in Books, Kids, pictures, Statistics, Travel, University life with tags AABI, coupling, discussion paper, Journal of the Royal Statistical Society, Markov chain Monte Carlo algorithm, MCMC, Read paper, Royal Statistical Society, Series B, unbiasedness, Université Paris Dauphine, Vancouver on December 10, 2019 by xi'an**T**he paper ‘Unbiased Markov chain Monte Carlo methods with couplings’ by Pierre Jacob et al. will be discussed (or Read) tomorrow at the Royal Statistical Society, 12 Errol Street, London, tomorrow night, Wed 11 December, at 5pm London time. With a pre-discussion session at 3pm, involving Chris Sherlock and Pierre Jacob, and chaired by Ioanna Manolopoulou. While I will alas miss this opportunity, due to my trip to Vancouver over the weekend, it is great that that the young tradition of pre-discussion sessions has been rekindled as it helps put the paper into perspective for a wider audience and thus makes the more formal Read Paper session more profitable. As we discussed the paper in Paris Dauphine with our graduate students a few weeks ago, we will for certain send one or several written discussions to Series B!

## unbiased estimators that do not exist

Posted in Statistics with tags binomial distribution, counterexample, cross validated, exam, existence of unbiased estimators, monomial, teaching, unbiasedness on January 21, 2019 by xi'an**W**hen looking at questions on X validated, I came across this seemingly obvious request for an unbiased estimator of **P**(X=k), when X~**B**(n,p). Except that X is not observed but only Y~**B**(s,p) with s<n. Since **P**(X=k) is a polynomial in p, I was expecting such an unbiased estimator to exist. But it does not, for the reasons that Y only takes s+1 values and that any function of Y, including the MLE of **P**(X=k), has an expectation involving monomials in p of power s at most. It is actually straightforward to establish properly that the unbiased estimator does not exist. But this remains an interesting additional example of the rarity of the existence of unbiased estimators, to be saved until a future mathematical statistics exam!

## the [not so infamous] arithmetic mean estimator

Posted in Books, Statistics with tags arithmetic mean estimator, Bayesian Analysis, Chib's approximation, harmonic mean estimator, HPD region, importance sampling, label switching, mixture of distributions, nested sampling, unbiasedness on June 15, 2018 by xi'an

“Unfortunately, no perfect solution exists.”Anna Pajor

**A**nother paper about harmonic and not-so-harmonic mean estimators that I (also) missed came out last year in Bayesian Analysis. The author is Anna Pajor, whose earlier note with Osiewalski I also spotted on the same day. The idea behind the approach [which belongs to the branch of Monte Carlo methods requiring additional simulations after an MCMC run] is to start as the corrected harmonic mean estimator on a restricted set **A** as to avoid tails of the distributions and the connected infinite variance issues that plague the harmonic mean estimator (an old ‘Og tune!). The marginal density p(y) then satisfies an identity involving the prior expectation of the likelihood function restricted to **A** divided by the posterior coverage of **A**. Which makes the resulting estimator unbiased only when this posterior coverage of **A** is known, which does not seem realist or efficient, except if **A** is an HPD region, as suggested in our earlier “safe” harmonic mean paper. And efficient only when **A** is well-chosen in terms of the likelihood function. In practice, the author notes that P(**A**|y) is to be estimated from the MCMC sequence and that the set **A** should be chosen to return large values of the likelihood, p(y|θ), through importance sampling, hence missing somehow the double opportunity of using an HPD region. Hence using the same default choice as in Lenk (2009), an HPD region which lower bound is derived as the minimum likelihood in the MCMC sample, “range of the posterior sampler output”. Meaning P(**A**|y)=1. (As an aside, the paper does not produce optimality properties or even heuristics towards efficiently choosing the various parameters to be calibrated in the algorithm, like the set **A** itself. As another aside, the paper concludes with a simulation study on an AR(p) model where the marginal may be obtained in closed form if stationarity is not imposed, which I first balked at, before realising that even in this setting both the posterior and the marginal do exist for a finite sample size, and hence the later can be estimated consistently by Monte Carlo methods.) A last remark is that computing costs are not discussed in the comparison of methods.

The final experiment in the paper is aiming at the marginal of a mixture model posterior, operating on the galaxy benchmark used by Roeder (1990) and about every other paper on mixtures since then (incl. ours). The prior is pseudo-conjugate, as in Chib (1995). And label-switching is handled by a random permutation of indices at each iteration. Which may not be enough to fight the attraction of the current mode on a Gibbs sampler and hence does not automatically correct Chib’s solution. As shown in Table 7 by the divergence with Radford Neal’s (1999) computations of the marginals, which happen to be quite close to the approximation proposed by the author. (As an aside, the paper mentions poor performances of Chib’s method when centred at the posterior mean, but this is a setting where the posterior mean is meaningless because of the permutation invariance. As another, I do not understand how the RMSE can be computed in this real data situation.) The comparison is limited to Chib’s method and a few versions of arithmetic and harmonic means. Missing nested sampling (Skilling, 2006; Chopin and X, 2011), and attuned importance sampling as in Berkoff et al. (2003), Marin, Mengersen and X (2005), and the most recent Lee and X (2016) in Bayesian Analysis.

## unbiased consistent nested sampling via sequential Monte Carlo [a reply]

Posted in pictures, Statistics, Travel with tags auxiliary variable, Brisbane, evidence, marginal likelihood, nested sampling, Og, particle filter, QUT, unbiasedness on June 13, 2018 by xi'an*Rob Salomone sent me the following reply on my comments of yesterday about their recently arXived paper.*

“Which never occurred as the number one difficulty there, as the simplest implementation runs a Markov chain from the last removed entry, independently from the remaining entries. Even stationarity is not an issue sinceI believe that the first occurrence within the level set is distributed from the constrained prior.”

“And then, in a twist that is not clearly explained in the paper, the focus moves to an improved nested sampler that moves one likelihood value at a time, with a particle step replacing a singleparticle. (Things get complicated when several particles may take the very same likelihood value, but randomisation helps.) At this stage the algorithm is quite similar to the original nested sampler. Except for the unbiased estimation of the constants, thefinal constant, and the replacement of exponential weights exp(-t/N) by powers of (N-1/N)”

**is**a special case of SMC (with the weights replaced with a suboptimal choice).